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Full text of "Treatise On Analysis Vol-Ii"

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constant y > 0 such that

(                        V(x\y) = (Ufa) \Ufry)

(for the scalar product defined in (15.7.4)).

If A is finite-dimensional, then the image of A under Ul is the algebra
Endc(I) of all endomorphisms of the vector space I, and the relation (
remains valid, for the scalar product defined in ( (from the scalar
product on I -which is the restriction of that on A).

We can assume that A is the Hilbert sum of a (finite or infinite) sequence of
minimal left ideals ! = Aen, where I = Ix ( and all the In are isomor-
phic (15.8.13). If x T 0 and Ufa) = 0, that is, if xl = {0}, we should have
(Ax)l = {0}, and since the ideal Ax is nonzero, it contains a minimal left ideal
I7 (15.8.8), which must be isomorphic to I (15.8.13); hence I'l = {0}, contrary
to (15.8.12(iii)). Hence the representation /r is faithful. Put Pn = t/i(en),
which is the orthogonal projection of I = Ael onto the one-dimensional subspace
en A*i (15.8.12), because (xe1 - enxel \ enyej = (enxei - el xel \ye^ = 0.
Since emen = 0 if m ^ n, we have PmPn=0 and therefore the subspaces
enAei are orthogonal in pairs. Moreover, I is the Hilbert sum of the sub-
spaces enAel9 because, if xe1 is orthogonal to all these subspaces, we have
Pn(xe1) = 0 for all n, so that enxe = 0 for all , and therefore xe1 belongs
to the right annihilator of A, which is zero ( This shows that the
sequence (ln) is finite if and only if I (and therefore each In) is finite-dimQn-
sional over C, or equivalently if and only if A is finite-dimensional.

Let (an) be an orthonormal basis of I such that an e en Ael for each n.
Then anafeenAen, hence ana* = !<? for some AB e C* (15.8.12). Likewise
We have ^ = An, because on the one hand

and on the other hand

! anan a* = % *n el<*n   = *>'n ^n * * = ^ An en ,

since an et = an. Also

1 = fa, I flj = (a,, I en an) = (ana* \ en) = Xn(en \ en)

1 = (fl. IO = (^ I a, cx) = (a*aM | ej = An(^ | ej;

(                       feU) = fc|e,)ce also contains